calculate the 16th term of an arithmatic progression he its fifth term is 6 and its 12th is 41.
a+4d = 6
a+11d = 41
7d = 35
d = 5
a = -14
-14 + 15(5) = 61
To find the 16th term of an arithmetic progression, we can use the formula:
an = a1 + (n - 1)d
where:
an is the nth term,
a1 is the first term, and
d is the common difference.
First, we need to find the value of a1 (first term) and d (common difference).
Given that the fifth term (a5) is 6 and the twelfth term (a12) is 41, we can set up the following equations:
a5 = a1 + 4d
a12 = a1 + 11d
Since the fifth term is given as 6, we can substitute this into the first equation:
6 = a1 + 4d Equation 1
Similarly, we can substitute the twelfth term of 41 into the second equation:
41 = a1 + 11d Equation 2
Now we have a system of two equations with two variables. We can solve this system to find the values of a1 and d.
First, let's solve Equation 1 for a1:
a1 = 6 - 4d
Now substitute this value of a1 into Equation 2:
41 = (6 - 4d) + 11d
Simplifying the equation:
41 = 6 + 7d
Subtract 6 from both sides:
35 = 7d
Divide both sides by 7:
d = 5
Now we have the value of d, which is the common difference. We can substitute this back into Equation 1 to find a1:
6 = a1 + 4(5)
6 = a1 + 20
Subtract 20 from both sides:
a1 = -14
Now that we have the values of a1 and d, we can use the formula to find the 16th term (a16):
a16 = a1 + (16 - 1)d
a16 = -14 + 15(5)
a16 = -14 + 75
a16 = 61
Therefore, the 16th term of the arithmetic progression is 61.